Harnessing stellar kinematics to constrain dark energy with the double-source-plane gravitational lens SDSS J0946+1006

SDSS J0946+1006 is an attractive target for measuring cosmological parameters through lens modelling. It is the best-studied galaxy-scale strong gravitational lens with multiple sources whose redshift separations are well-suited to constraining the dark energy equation of state. However, multi-plane lens models with free cosmological parameters risk a multi-plane mass-sheet degeneracy, although this can be lifted by a non-lensing deflector density profile tracer. We simultaneously reconstruct near-infrared and near-ultraviolet HST imaging whilst including a velocity dispersion measurement from VLT-MUSE to constrain the foreground deflector. Imaging is reconstructed faithfully regardless of whether the inferred kinematics are realistic, though we find the kinematic constraint essential for shifting the preferred cosmology into a region not in significant tension with other dark energy probes. Deflector density profile perturbations, via substructure and multipolar halo shape deformations, have only a modest effect on inferred cosmology. Combining our fiducial model with Planck CMB data yields w=-1.01+0.08-0.13; or, combined with Pantheon SNe Ia, w=-0.99+0.13-0.15. We further show that this system proves a remarkably valuable complementary probe in the w 0 - w a plane of an evolving dark energy model, and that the DESI BAO tension with Λ CDM seen when combined with other datasets is not reproduced when combined with this lens, yielding (w 0, w a)=(-0.87+0.10-0.11, -0.22+0.28-0.25). The system's third source, visible with MUSE, only weakly constrains w CDM but may strengthen w 0w a CDM constraints, though further mass-model complexity along its line of sight is required. Overall, kinematics-informed multi-plane lens modelling is a robust route to competitive dark energy constraints, even with a single system.